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S T kF R(x), L(x) (x) = R+L R RR L LL, -iRxR = R2 iLxL = L2 i [R(x), R(y)] = -[L(x), L(y)] = x(x-y) 2 22 R (ix)2 R = R3. 2m 3m RR q q/(vF q - ) vF q = (q, ) m Im = (1 - ||) 2q = ( - qvF )/wq wq = q2/(2m) (x) = 1 x > 0 1/m (q, ) g2RL m-1(R3 + L3) u - v R, L 1 2 H = v R2 + L2 + (R3 + L3) + (R2L + RL2), 3 2 2 3 + K2 2 (K2 - 1) 1 = , 2 = , m m 2 K K v K 1 - g2/(2vF ) K < 1 K = 1 1/m 1 R3 + L3 g2 vF q4 + vq - vq Im q coth + coth, 2 2kF (2 - q2v2) 4T 4T 4vF q4T >, T ||, |qv| 2 kF (2 - q2v2)2 = K(K2 - 1)2/64 u12 000 111 000 111 000 111 00000011111100000000111111111 1 1 0000001111110001110001110000001111110000000000 0001111111111 11100000011111100000 00011111 1110000001111110001110001110000001111112 2 2 00011100011111 = + + 00000011111100011100000000 00011111111 11100000000111111110100011101000111 3 3 3 01 T T1 = kF (v1 - v2)2 + u2 v1, kF = mvF |v1 - v2| u12 T < T1 (v1 - v2)kF 5 T U12(1 - K)R12 2 2 vF EF T1 vF v1 = v2 T < T U12kF T R12 vF EF T 1/U12 T > T T1 2 T1 T T > T1 R12 vF EF T = T1 T T R12 T R12 T g2 G T G 1 - (T0/T )2(1-K) -G (T/T0)2(K -1) 1 K = K = 2 S L H = vF dx R(ix)R - L(ix)L + g2 dx (RR)(LL), - -L - |x| < a kF g2 |x| > a a t = t = cos r = - = i sin ei r k(x) = (eikx + re-ikx)c + te-ikxc, x < 1k 2k = teikxc + (ikx + e-ikx)c, x > 1k 2k c c 1k 2k J0 + J3 J1 - iJ2 c(x)c1(x) c(x)c2(-x) 1 J1 + iJ2 J0 - J3 c(-x)c1(x) c(-x)c2(-x) 2 J0 J = (J1, J2, J3) U(1) SU(2) 0 2 2 2(x)) H = 2vF dx (J0(x) + J3 (x)) + 2vF dx (J0(x) + J- L + 2g2 dx (J0(-x)J0(x) - J3(-x)J3(x)) a J3 = (RJ)3 J R = T r[.S..S].

J t = 0 J3 = -J3 U(1) SU(2) g2 g1 1 gJ g2 J g 2 2 g2 7! = 5040 T = 0 = ln(L/L0) = ln(T/T0) T vF /L L L0 T0 0 < G = cos2 < 1 Y = Yr - Y Y = 2G - 1 = cos 2 g Yr Yr = Y0 + b11g + b22g22 + b21g2 + b33g33 + b32g32 + b31g3 +... bij Y (g)n -x -x -x -z -z -z (a) (b) (c) z z z y y y g2 gnm n > m Y0 Y0 Yr( = 0) = Y + b20g2 + b30g3 +...

g = 0 Yr(g, Y, T ) = F (g, T/), = (g, Y ) L T = 0 Yr(g,Y,) = (g, Yr()) (g, Yr) = dYr(g, ) = 1(g, Y ) = (g, Y0) d = Y Y0 Y0 Y 1(g, Y ) (g, Y0) = 0 1 Yr() = Y0 + (Y0) + 2(Y0) (Y0) + 3(Y0)[(Y0) (Y0)] +... 2 Y g Y0 - Y g Y Y + 2 (Y ) = (1 - Y ) -g + g2 - g3 + c3g3(1 - Y )2 + O(g4) 2 2 ln 2 1 2 c3 = + - 0.67 T = 0 c3 = + ln 2 - 0.12 2 2 24 T vF /L -x1 -x1 -x1 -x2 -x1 -z -x2 -x1 -z -x= + + + + x2 x1 x1 x2 x1 z x2 x1 z x ln(T0/T ) dY 2g(1 - Y ) - = - c3g3(1 - Y )2(1 + O(gY )) d 1 + 1 - g2 + gY |Y | 1 G = (1 + Y )/2 (T/T0)2(1-K) = (G)/(G0), GK (G) = (K + G(1 - K))4c (1-K), 1 - G G = G0 T = T0 c3 c3 0.60 T vF /L < T vF /L G 1 G 0 K 1 c3 G 1/2 c3 c3 = 1/4 S 4kF 1 1 m2 L = (t)2 - (x)2 + cos + H x, 2 2 2 2 H K = 2/(4) = 8 4kF H = 0 H = 0 = L-1 L dx = x() 0 2 K = 2/(4) 1 2/(8) 8m/2 m 0 = 0 + (q) = (m/k) dn() q = 2(i(K/)Z() + 1/2) K dn() Z(u) = d ln 4(u/(2K))/du k m/2 = kK(k) 3.

(a) 2.2.2.

1.1.1.

0.0.-2 -1 0 1 2 - 0.5 0 0.5 1.

q/m q/ /m = 0.26 /m = 0. 1 2 3.

0(x) 0 = 0 x + -1 = 2 /m q /m ( ) K = 2/(16 1 - k2K2), k K m k = 0 K = 2/(4) /2 4kF 2n = 0 D = 2/(32EK) D 0 K XXZ 2 3 6+x c j + 1/2 j - 1/ j + 1/2 j j j - 1/2 j N H1 = - (c cj+1/2 + h.c.).

j+1+1/j= J j +1/2 j j + 1 J N+Nh- H2 = Ji,i+1i.i+1, i= N + Nh N(1 + c) i Ji,i+1 = J J (i, i + 1) J /J 0.75 J = J J = J i N(1 + c) l- l nj+1/2 j= nj+1/2 = c cj+1/2 j+1/ l-l+ nj+1/2 k j= l- Sl = (N(1 + c))-1/2 k exp ik(l + nj+1/2), k j= H1 (q, ) = SS q, (1 + c) S1,2 D(S1 S2) D D JSlSl+1 J D/J 0.1 D H e-iUHeiU L l- U = lSlz, l = tan-1(Di/J), l=1 i= D eiU zz zz(q, ) xx, yy Dl = J tan (q, ) xx, xy 1 1, i 1, -i = G(q + , ) + G(q - , ) yx, yy 2 -i, 1 i, G(q, ) (q, ) d21 G(q + , ) - G(q - , ) (DQ)(QP0)Im.

dd 2D Q = Q/Q Q P0 q Q 2 3 6+x 2 3 6+x 2 3 6+x S = 1/ 2 3 6+x S = 1/2

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